Theorem pm110.643 6072

Description: 1+1=2 for cardinal number addition. Theorem *110.643 of Principia Mathematica, vol. II, p. 86, which adds the remark, "The above proposition is occasionally useful." Unlike us, Whitehead and Russell define cardinal addition on collections of all sets equinumerous to 1 and 2 (which for us are proper classes unless we restrict them as in karden 5856), but after applying definitions, our theorem is equivalent. See also the comment for pm54.43 5662. The comment for cdavali 6068 explains why we use ~~ instead of =.

Assertion

Ref	Expression
pm110.643	|- (1o +c 1o) ~~ 2o

Proof of Theorem pm110.643

Step	Hyp	Ref	Expression
1		1on 5182	. . . 4 |- 1o e. On
2	1	elisseti 2301	. . 3 |- 1o e. _V
3	2, 2	cdavali 6068	. 2 |- (1o +c 1o) = ((1o X. {(/)}) u. (1o X. {1o}))
4		xp01disj 5188	. . 3 |- ((1o X. {(/)}) i^i (1o X. {1o})) = (/)
5		0ex 3446	. . . . 5 |- (/) e. _V
6	2, 5	xpsnen 5494	. . . 4 |- (1o X. {(/)}) ~~ 1o
7	2, 2	xpsnen 5494	. . . 4 |- (1o X. {1o}) ~~ 1o
8		pm54.43 5662	. . . 4 |- (((1o X. {(/)}) ~~ 1o /\ (1o X. {1o}) ~~ 1o) -> (((1o X. {(/)}) i^i (1o X. {1o})) = (/) <-> ((1o X. {(/)}) u. (1o X. {1o})) ~~ 2o))
9	6, 7, 8	mp2an 761	. . 3 |- (((1o X. {(/)}) i^i (1o X. {1o})) = (/) <-> ((1o X. {(/)}) u. (1o X. {1o})) ~~ 2o)
10	4, 9	mpbi 206	. 2 |- ((1o X. {(/)}) u. (1o X. {1o})) ~~ 2o
11	3, 10	eqbrtri 3356	1 |- (1o +c 1o) ~~ 2o

Syntax hints:   <-> wb 163   = wceq 1298   u. cun 2591   i^i cin 2592  (/)c0 2875  {csn 3044   class class class wbr 3338  Oncon0 3657   X. cxp 3984  (class class class)co 4884  1oc1o 5172  2oc2o 5173   ~~ cen 5423   +c ccda 6065

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1o 5177  df-2o 5178  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-cda 6066

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